<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#NM1" title="GLIB_003:NM.1">WGraph</a>;<br/><b>let </b><font color="Maroon" title="c2">src</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>set </b><font color="Maroon" title="c3">DCS</font> =  <a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> $1</span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>;<br/>
<a NAME="E1:33"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K4" title="GLIB_004:func.4">DIJK:Init</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font>
 
<b>by </b><i><a class="ref" href="glib_004.html#D7" target="_self" title="GLIB_004:def.7">Def7</a></i>;<br/>
<a NAME="E2:33"/><b>then </b><i><font color="Green" title="E22">A1</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> 0</span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">src</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="glib_004.html#T15" target="_self" title="GLIB_004:th.15">Th15</a></i>;<br/>
<a NAME="E3:33"/>
<font color="Maroon" title="c2">src</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 
<b>by </b><i><a class="ref" href="glib_002.html#T18" title="GLIB_002:th.18">GLIB_002:18</a></i>;<br/>
<a NAME="E4:33"/><b>then </b><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="glib_004.html#E2:33"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:33_1"/><i><font color="Green" title="E24">A3</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 ;<br/><b>set </b><font color="Maroon" title="c5">Gk</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font>;<br/><b>set </b><font color="Maroon" title="c6">NextG</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon" title="c7">VGk</font> =  <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c8">VNextG</font> =  <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c9">BestEdges</font> =  <a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c10">e</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c11">NextEG</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span>,1;<br/><b>set </b><font color="Maroon" title="c12">v1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c13">target</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c14">pc</font> = <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c15">ec</font> = <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E2:33_1"/><i><font color="Green" title="E25">A4</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_004.html#D7" target="_self" title="GLIB_004:def.7">Def7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:33_1_1"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1"/>
(  <a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_1"/>
 <a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:33_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:33_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="glib_004.html#D5" target="_self" title="GLIB_004:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_1_1_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:33_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="glib_004.html#E1:33_1_1"><i><font color="Green" title="E26">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_2"/><i><font color="Green" title="E27">A6</font></i>: 
 <a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:33_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E28">A7</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13" title="GLIB_003:func.13">.labelVertex</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:33_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="glib_004.html#D5" target="_self" title="GLIB_004:def.5">Def5</a></i>;<br/><b>set </b><font color="Maroon" title="c17">xx</font> = <font color="Maroon" title="c16">x</font>;<br/><b>set </b><font color="Maroon" title="c18">GE</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13" title="GLIB_003:func.13">.labelVertex</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E3:33_1_1_1_1_2"/>
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <span class="p1">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_004.html#T17" target="_self" title="GLIB_004:th.17">Th17</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c19">xx</font> = <font color="Maroon" title="c16">x</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="glib_004.html#E1:33_1_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/><a NAME="E5:33_1_1_1_1_2"/>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span>,<span class="p1">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:33_1_1_1_1_2"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="glib_004.html#D4" target="_self" title="GLIB_004:def.4">Def4</a></i>;<br/><a NAME="E6:33_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E29">A8</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#D18" title="GLIB_000:def.18">GLIB_000:def 18</a></i>;<br/><a NAME="E7:33_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:33_1"><i><font color="Green" title="E24">A3</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c20">v1'</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c19">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  or  not <font color="Maroon" title="c19">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c19">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c19">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:33_1"><i><font color="Green" title="E24">A3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_2_1_1_2"/>
not <font color="Maroon" title="c19">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 ;<br/><div class="add"><a NAME="E2:33_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E30">A9</font></i>: 
not <font color="Maroon" title="c19">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="glib_003.html#T43" title="GLIB_003:th.43">GLIB_003:43</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><a NAME="E1:33_1_1_1_1_2_1_1_2_1"/><i><font color="Green" title="E31">A10</font></i>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:33_1_1_1_1_2"><i><font color="Green" title="E27">A6</font></i></a></i>;<br/><a NAME="E2:33_1_1_1_1_2_1_1_2_1"/><i><font color="Green" title="E32">A11</font></i>: 
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <span class="p1">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="ref" href="glib_004.html#T17" target="_self" title="GLIB_004:th.17">Th17</a></i>;<br/><a NAME="E3:33_1_1_1_1_2_1_1_2_1"/><b>then </b>
(  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_1_1_2_1_1_2_1"/>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:33_1_1_1_1_2_1_1_2_1"><i><font color="Green" title="E31">A10</font></i></a></i>;<br/><b>thus </b><a NAME="E5:33_1_1_1_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:33_1_1_1_1_2_1_1_2_1"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/><a NAME="E6:33_1_1_1_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">xx</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:33_1_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="glib_004.html#E2:33_1_1_1_1_2"><i><font color="Green" title="E28">A7</font></i></a>, <a class="txt" href="glib_004.html#E2:33_1_1_1_1_2_1_1_2"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="glib_003.html#T62" title="GLIB_003:th.62">GLIB_003:62</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_1_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">xx</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:33_1_1_1_1_2_1_1_2_1"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/></div><b>end;</b></div><a NAME="E4:33_1_1_1_1_2_1_1_2"/><b>then </b>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c20">v1'</font>,<font color="Maroon" title="c19">xx</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D16" title="GLIB_000:def.16">GLIB_000:def 16</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c19">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:33_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="glib_004.html#E6:33_1_1_1_1_2"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="glib_002.html#T19" title="GLIB_002:th.19">GLIB_002:19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:33_1_1_1_1_2"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:33"/><b>then </b><i><font color="Green" title="E24">A12</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
;<br/>
<a NAME="E7:33"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="glib_004.html#E4:33"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="glib_004.html#E6:33"><i><font color="Green" title="E24">A12</font></i></a>);<br/></i>
<b>hence </b><a NAME="E8:33"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 ;<br/>


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